2D Laplacian in polar coordinates

[Mayan Fung]

The 2D Laplacian in polar coordinates has the form of
$$ \frac{1}{r}(ru_r)_r +\frac{1}{r^2}u_{\theta \theta} =0 $$
By separation of variables, we can write the ## \theta## part as
$$ \Theta'' (\theta) = \lambda \Theta (\theta)$$

Now, the book said because we need to satisfy the condition ## \Theta (0)=\Theta (2\pi)##, ##\lambda = n^2## with n =1,2,3...

I tried this in detailed steps.
##\Theta (\theta) = Acos\sqrt{\lambda}\theta + Bcos\sqrt{\lambda}\theta##
With the BC,
## B = Acos(\sqrt{\lambda}2\pi) + Bsin(\sqrt{\lambda}2\pi) ##
This is just an equality but not an identity. Why can't I conclude that
## B = \frac{cos(\sqrt{\lambda}2\pi)}{1-sin(\sqrt{\lambda}2\pi)} A ##
instead of imposing constraint on ##\lambda##

Thanks!

 

[Orodruin]

Because you also need to satisfy ##\Theta'(0) = \Theta'(2\pi)##. For general lambda that gives you a system of equations that lead to trivial solutions only. Only for a discrete set of lambda do you get non-trivial solutions.

Note: You are missing the ##\lambda=0## solution.

 

[Mayan Fung]

Because you also need to satisfy ##\Theta'(0) = \Theta'(2\pi)##. For general lambda that gives you a system of equations that lead to trivial solutions only. Only for a discrete set of lambda do you get non-trivial solutions.

Note: You are missing the ##\lambda=0## solution.

Thank you! Is it also true that ##\Theta^{(n)}(0)=\Theta^{(n)}(2\pi)##?

 

[Orodruin]

Thank you! Is it also true that ##\Theta^{(n)}(0)=\Theta^{(n)}(2\pi)##?

That follows from the differential equation if you have the other two.